trig-atan-01 true

category inverse

Summary

Given

\(x \in \mathbb{R},\quad x > -1,\quad x < 1\)

Claim

\(\displaystyle \arctan\left(\tan x\right) = x\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

Answers (latest run per system; open code for the reproducer that ran)

system outcome answer steps (from the trace) s run query (as sent) detail

none

always-none no verdict null 0.0 2026-10-13 17:21:03 7a1dcfa636ed (real x) ∧ (> x -1) ∧ (< x 1) ⊢ (= (atan (tan x)) x) null

sympy

sympy-ask no verdict null difference → -x + atan(tan(x)) · ask → None 0.109 2026-10-13 17:21:03 7a1dcfa636ed Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), x > -1, x < 1)
Residual = atan(tan(x)) - x
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → -x + atan(tan(x)) · ask → None 0.03 2026-10-12 20:21:44 45d6b9056f06 Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), x > -1, x < 1)
Residual = atan(tan(x)) - x
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine no verdict null difference → -x + atan(tan(x)) · refine → -x + atan(tan(x)) 0.038 2026-10-13 17:21:06 7a1dcfa636ed Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), x > -1, x < 1)
Residual = atan(tan(x)) - x
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
refine	`refine(Residual, Assum)`	→	`-x + atan(tan(x))`
answer			`null`

sympy-refine no verdict null difference → -x + atan(tan(x)) · refine → -x + atan(tan(x)) 0.041 2026-10-12 20:21:45 45d6b9056f06 Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), x > -1, x < 1)
Residual = atan(tan(x)) - x
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
refine	`refine(Residual, Assum)`	→	`-x + atan(tan(x))`
answer			`null`

sympy-simplify no verdict null difference → -x + atan(tan(x)) · simplify → -x + atan(tan(x)) 0.044 2026-10-13 17:21:07 7a1dcfa636ed Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
# dropped (not a unary fact about one symbol): x > -1
# dropped (not a unary fact about one symbol): x < 1
Residual = atan(tan(x)) - x
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
simplify	`simplify(Residual)`	→	`-x + atan(tan(x))`
answer			`null`

sympy-simplify no verdict null difference → -x + atan(tan(x)) · simplify → -x + atan(tan(x)) 0.044 2026-10-12 20:21:47 45d6b9056f06 Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
# dropped (not a unary fact about one symbol): x > -1
# dropped (not a unary fact about one symbol): x < 1
Residual = atan(tan(x)) - x
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
simplify	`simplify(Residual)`	→	`-x + atan(tan(x))`
answer			`null`

wolfram

wolfram-fullsimplify solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-13 17:21:15 7a1dcfa636ed Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-13 17:21:12 7a1dcfa636ed Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-13 17:21:08 7a1dcfa636ed Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Older runs (11)

system outcome answer steps (from the trace) s run query (as sent) detail

wolfram-fullsimplify solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-13 16:39:24 1e4dbd2f7d86 Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-13 16:39:21 1e4dbd2f7d86 Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-13 16:39:18 1e4dbd2f7d86 Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

sympy-simplify no verdict null difference → -x + atan(tan(x)) · simplify → -x + atan(tan(x)) 0.044 2026-10-13 16:39:17 1e4dbd2f7d86 Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
# dropped (not a unary fact about one symbol): x > -1
# dropped (not a unary fact about one symbol): x < 1
Residual = atan(tan(x)) - x
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
simplify	`simplify(Residual)`	→	`-x + atan(tan(x))`
answer			`null`

sympy-refine no verdict null difference → -x + atan(tan(x)) · refine → -x + atan(tan(x)) 0.038 2026-10-13 16:39:16 1e4dbd2f7d86 Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), x > -1, x < 1)
Residual = atan(tan(x)) - x
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
refine	`refine(Residual, Assum)`	→	`-x + atan(tan(x))`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:13 1e4dbd2f7d86 (real x) ∧ (> x -1) ∧ (< x 1) ⊢ (= (atan (tan x)) x) null

sympy-ask no verdict null difference → -x + atan(tan(x)) · ask → None 0.109 2026-10-13 16:39:13 1e4dbd2f7d86 Q.real(x) ∧ (x > (-1)) ∧ (x < 1) ⊢ Eq(atan(tan(x)), x) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), x > -1, x < 1)
Residual = atan(tan(x)) - x
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`atan(tan(x)) - x`	→	`-x + atan(tan(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-12 20:21:54 45d6b9056f06 Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-12 20:21:51 45d6b9056f06 Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

always-none no verdict null 0.0 2026-10-12 20:21:48 45d6b9056f06 (real x) ∧ (> x -1) ∧ (< x 1) ⊢ (= (atan (tan x)) x) null

wolfram-simplify solved true difference → -x + ArcTan[Tan[x]] · result → 0 0.022 2026-10-12 20:21:48 45d6b9056f06 Element[x, Reals] ∧ (x > (-1)) ∧ (x < 1) ⊢ (ArcTan[Tan[x]] == x) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((x > (-1))) && ((x < 1));
  Residual = (ArcTan[Tan[x]]) - (x);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTan[Tan[x]]) - (x)`	→	`-x + ArcTan[Tan[x]]`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Source (trig.sexp)

(case trig-atan-01 (category inverse)
  (var x real)
  (assume (> x -1)) (assume (< x 1))
  (ask (= (atan (tan x)) x))
  (expect true))